Quasi-stationary and ratio of expectations distributions: A comparative study

Many stochastic systems, including biological applications, use Markov chains in which there is a set of absorbing states. It is then needed to consider analogs of the stationary distribution of an irreducible chain. In this context, quasi-stationary distributions play a fundamental role to describe the long-term behavior of the system. The rationale for using quasi-stationary distribution is well established in the abundant existing literature. The aim of this study is to reformulate the ratio of means approach ( [Darroch and Seneta, 1965] and [Darroch and Seneta, 1967]) which provides a simple alternative. We have a two-fold objective. The first objective is viewing quasi-stationarity and ratio of expectations as two different approaches for understanding the dynamics of the system before absorption. At this point, we remark that the quasi-stationary distribution and a ratio of means distribution may give or not give similar information. In this way, we arrive to the second objective; namely, to investigate the possibility of using the ratio of expectations distribution as an approximation to the quasi-stationary distribution. This second objective is explored by comparing both distributions in some selected scenarios, which are mainly inspired in stochastic epidemic models. Previously, the rate of convergence to the quasi-stationary regime is taking into account in order to make meaningful the comparison.     

A novel fitness proxy in structured locally finite metapopulations with diploid genetics, with an application to dispersal evolution

Many studies of evolutionarily stable strategies (ESS) for technical reasons make the simplification that reproduction is clonal. A post-hoc justification is that in the simplest eco-evolutionary models more realistic genetic assumptions, such as haploid sexual or diploid sexual cases, yield results compatible with the clonal ones. For metapopulations the technical reasons were even more poignant thanks to the lack of accessible fitness proxies for the diploid case. However, metapopulations are also precisely the sort of ecological backdrop for which one expect discrepancies between the evolutionary outcomes derived from clonal reproduction and diploid genetics, because substantially many mutant homozygotes appear locally even though the mutant is rare globally. In this paper we devise a fitness proxy applicable to the haploid sexual and diploid sexual case, in the style of Metz and Gyllenberg [Metz, J.A.J., Gyllenberg, M., 2001. How should we define fitness in structured metapopulation models? Including an application to the calculation of ES dispersal strategies. Proc. R. Soc. Lond. B 268, 499-508], that can cope with local population fluctuations due to environmental and demographic stochasticity. With the use of this fitness proxy we find that in dispersal evolution the studied clonal model is equivalent with the haploid sexual model, and that there are indeed many differences between clonal and diploid ESS dispersal rates. In a homogenous landscape the discrepancy is but minor (less than 2%), but the situation is different in a heterogeneous landscape: Not only is the quantitative discrepancy between the two types of ESSs appreciable (around 10%-20%), but more importantly, at the same parameter values, evolutionarily stability properties may differ. It is possible, that the singular strategy is evolutionarily stable in the clonal case but not in the diploid case, and vice versa.     

A generalization of Gompertz law compatible with the Gyllenberg-Webb theory for tumour growth

We present a new extension of Gompertz law for tumour growth and anti-tumour therapy. After discussing its qualitative and analytical properties, we show, in the spirit of [16], that, like the standard Gompertz model, it is fully compatible with the two-population model of Gyllenberg and Webb, formulated in [14] in order to provide a theoretical basis to Gompertz law. Compatibility with the model proposed in [17] is also investigated. Comparisons with some experimental data confirm the practical applicability of the model. Numerical simulations about the method performance are presented.     

Stabilizing Spatially-Structured Populations through Adaptive Limiter Control

Stabilizing the dynamics of complex, non-linear systems is a major concern across several scientific disciplines including ecology and conservation biology. Unfortunately, most methods proposed to reduce the fluctuations in chaotic systems are not applicable to real, biological populations. This is because such methods typically require detailed knowledge of system specific parameters and the ability to manipulate them in real time; conditions often not met by most real populations. Moreover, real populations are often noisy and extinction-prone, which can sometimes render such methods ineffective. Here, we investigate a control strategy, which works by perturbing the population size, and is robust to reasonable amounts of noise and extinction probability. This strategy, called the Adaptive Limiter Control (ALC), has been previously shown to increase constancy and persistence of laboratory populations and metapopulations of Drosophila melanogaster. Here, we present a detailed numerical investigation of the effects of ALC on the fluctuations and persistence of metapopulations. We show that at high migration rates, application of ALC does not require a priori information about the population growth rates. We also show that ALC can stabilize metapopulations even when applied to as low as one-tenth of the total number of subpopulations. Moreover, ALC is effective even when the subpopulations have high extinction rates: conditions under which another control algorithm had previously failed to attain stability. Importantly, ALC not only reduces the fluctuation in metapopulation sizes, but also the global extinction probability. Finally, the method is robust to moderate levels of noise in the dynamics and the carrying capacity of the environment. These results, coupled with our earlier empirical findings, establish ALC to be a strong candidate for stabilizing real biological metapopulations.     

The genetic balance between varying population size and selective neutrality

This note extends to an arbitrary offspring distribution the generalized model for random fluctuation of allele frequency, where population size is permitted to fluctuate randomly from generation to generation. Martingale methods analogous to those of Seneta (1974) and Heyde and Seneta (1975) are applied to discuss conditions for Pr(Y(1–Y)>0)>0, where Y is the (almost sure) limiting frequency of one allele. An overlapping generation study of the same phenomenon has recently been made by Heyde (1981).     

Stability aware spatial cut of metapopulations ecological networks

Ecological complex networks are common in the study of patched ecological systems where evolving populations interact within and among the patches. The loss of the dispersal connections between patches due to reasons such as erosion of migration corridors and road construction can cause an undesirable partitioning of such networks resulting in instability or negative impact on the metapopulations. A partitioning or spatial cut that is aware of the stability of the dynamics in the resulting daughter sub-networks can be an effective tool in dealing with the situation like proposing road alignment through a metapopulations network. This paper provides some mathematical conditions along with an heuristic graph partitioning algorithm that can help in finding ecologically suitable partitions of the metapopulations networks. Our study noted the crucial role of network connectivity (measured by Fiedler value) in stabilizing the metapopulations. That is, a sufficiently connected metapopulations network along with constrained internal patch dynamics has stable dynamics around its homogeneous co-existential equilibrium solution. With the considered mathematical model in this paper, network partitioning does not alter the internal patch dynamics around its homogeneous equilibrium point, but it can change the connectivity levels in the partitioned subnetworks. Thus, the proposed partitioning problem for an already stable metapopulations network is reduced to finding its subnetworks with desirable connectivity levels.     

Evolution of body condition‐dependent dispersal in metapopulations

Body condition‐dependent dispersal strategies are common in nature. Although it is obvious that environmental constraints may induce a positive relationship between body condition and dispersal, it is not clear whether positive body conditional dispersal strategies may evolve as a strategy in metapopulations. We have developed an individual‐based simulation model to investigate how body condition–dispersal reaction norms evolve in metapopulations that are characterized by different levels of environmental stochasticity and dispersal mortality. In the model, body condition is related to fecundity and determined either by environmental conditions during juvenile development (adult dispersal) or by those experienced by the mother (natal dispersal). Evolutionarily stable reaction norms strongly depend on metapopulation conditions: positive body condition dependency of dispersal evolved in metapopulation conditions with low levels of dispersal mortality and high levels of environmental stochasticity. Negative body condition‐dependent dispersal evolved in metapopulations with high dispersal mortality and low environmental stochasticity. The latter strategy is responsible for higher dispersal rates under kin competition when dispersal decisions are based on body condition reached at the adult life stage. The evolution of both positive and negative body condition‐dependent dispersal strategies is consequently likely in metapopulations and depends on the prevalent environmental conditions.     

Species delimitation and geography

Despite the importance of the geographical arrangement of populations for the inference of species boundaries, only a few approaches that integrate spatial information into species delimitation have thus far been developed. Persistent differentiation of sympatric groups of individuals is the best criterion for species status. Species delimitation becomes more prone to error if allopatric metapopulations are considered because it is often difficult to assess whether observed differences between allopatric metapopulations would be sufficient to prevent the fusion of these metapopulations upon contact. We propose a novel approach for testing the hypothesis that the multilocus genetic distances between individuals or populations belonging to two different candidate species are not larger than expected based on their geographical distances and the relationship of genetic and geographical distances within the candidate species. A rejection of this null hypothesis is an argument for classifying the two studied candidate species as distinct species. Case studies show that the proposed tests are suitable to distinguish between intra‐ and interspecific differentiation. The regression approach proposed here is more appropriate for testing species hypotheses with regard to isolation by distance than (partial) Mantel tests. Our tests assume a linear relationship between genetic and (transformed) geographical distances. This assumption can be compromised by a high genetic variability within populations as found in a case study with microsatellite markers.     

不同迁移能力和竞争能力的集合种群竞争模式及其模拟

种群竞争是影响生态系统演化的莺要生态过程之一。本文是在徐彩琳和Tilman研究工作的基础上,将竞争系数引入集合种群动力模式,建立了集合种群之间竞争的数学模型,并对5-集合种群的竞争动态进行了计算机模拟研究。结果表明：种群竞争排除与共存受迁移扩散能力和竞争能力影响很大,排除原理在理论上是存在的,在广域集合种群和实际中物种是竞争共存的．共存的条件是其竞争能力与扩散能力呈非线性负相关关系,竞争的结果使物种的强弱序列发生变化。     

Patch-occupancy dynamics in fragmented landscapes

Recent work on the dynamics of species living In fragmented landscapes has produced much Information on patterns of habitat patch occupancy in a wide range of organisms. Building on an elementary Markov chain model of patch occupancy, a family of Incidence-function models can be constructed for particular kinds of metapopulations. These models can be parameterized with field data on patch occupancy, and the models can be used to make quantitative predictions about specific metapopulations. This approach provides a potentially powerful tool for the management of reserve networks and species living in fragmented landscapes.     

RESISTANCE AND VIRULENCE STRUCTURE IN TWO LINUM MARGINALE-MELAMPSORA LINI HOST-PATHOGEN METAPOPULATIONS WITH DIFFERENT MATING SYSTEMS

Different patterns of resistance to six pathotypes of Melampsora lini were detected in 11 populations of Linum marginale distributed across two metapopulations. The two metapopulations (mountains and plains of New South Wales, Australia) differed in the annual cycle of disease development, which barely overlapped, and in the growth cycle and mating system of the host. Host populations in the mountains metapopulation were highly inbred, whereas those on the plains showed appreciable levels of outcrossing. Within each metapopulation there was significant variation among component populations in (1) levels of host resistance to individual pathogen isolates; (2) mean levels of resistance to all six isolates; (3) the number of resistance phenotypes present and the evenness of their distribution within the population; and (4) the average number of pathogen lines to which individual hosts were resistant. A more limited comparison of pathogen populations from the two metapopulations (two from each) found greater similarities in the structure of populations and particular virulence frequencies within, rather than among, the two metapopulations. Differences in host outcrossing rates between the two metapopulations are reflected in marked differences in the overall level of resistance, its partitioning within and among populations, the number and distribution of resistance phenotypes in the two areas, and the level of polymorphism for specific virulence factors in the pathogen, with the plains metapopulation showing consistently higher values. However, these differences were not significant. In general, variation for all parameters was just as great among populations within a metapopulation as between the two metapopulations.     

Dimorphism by Singularity Theory in a Model for River Ecology

Geritz, Gyllenberg, Jacobs, and Parvinen show that two similar species can coexist only if their strategies are in a sector of parameter space near a nondegenerate evolutionarily singular strategy. We show that the dimorphism region can be more general by using the unfolding theory of Wang and Golubitsky near a degenerate evolutionarily singular strategy. Specifically, we use a PDE model of river species as an example of this approach. Our finding shows that the dimorphism region can exhibit various different forms that are strikingly different from previously known results in adaptive dynamics.     

Refugia and connectivity sustain amphibian metapopulations afflicted by disease

Metapopulation persistence in fragmented landscapes depends on habitat patches that can support resilient local populations and sufficient connectivity between patches. Yet epidemiological theory for metapopulations has largely overlooked the capacity of particular patches to act as refuges from disease, and has suggested that connectivity can undermine persistence. Here, we show that relatively warm and saline wetlands are environmental refuges from chytridiomycosis for an endangered Australian frog, and act jointly with connectivity to sustain frog metapopulations. We coupled models of microclimate and infection probability to map chytrid prevalence, and demonstrate a strong negative relationship between chytrid prevalence and the persistence of frog populations. Simulations confirm that frog metapopulations are likely to go extinct when they lack environmental refuges from disease and lose connectivity between patches. This study demonstrates that environmental heterogeneity can mediate host–pathogen interactions in fragmented landscapes, and provides evidence that connectivity principally supports host metapopulations afflicted by facultative pathogens.     

Spatial scale of local adaptation in a plant-pathogen metapopulation

The rate and scale of gene flow can strongly affect patterns of local adaptation in host-parasite interactions. I used data on regional pathogen occurrence to infer the scale of pathogen dispersal and to identify pathogen metapopulations in the interaction between Plantago lanceolata and its specialist phytopathogen, Podosphaera plantaginis. Frequent extinctions and colonizations were recorded in the metapopulations, suggesting substantial gene flow at this spatial scale. The level of pathogen local adaptation was assessed in a laboratory inoculation experiment at three different scales: in sympatric host populations, in sympatric host metapopulations and in allopatric host metapopulations. I found evidence for adaptation to sympatric host populations, as well as evidence indicating that local adaptation may extend to the scale of the sympatric host metapopulation. There was also variation among the metapopulations in the degree of pathogen local adaptation. This may be explained by regional differences in the rate of migration.     

Assessing Extinction Risk in Small Metapopulations of Golden‐headed Lion Tamarins (Leontopithecus chrysomelas) in Bahia State,Brazil

Golden‐headed lion tamarins (GHLTs; Leontopithecus chrysomelas) are endangered primates endemic to the Brazilian Atlantic Forest, where loss of forest and its connectivity threaten species survival. Understanding the role of habitat availability and configuration on population declines is critical for guiding proactive conservation for this, and other, endangered species. We conducted population viability analysis to assess vulnerability of ten GHLT metapopulations to habitat loss and small population size. Seven metapopulations had a low risk of extirpation (or local extinction) over the next 100 years assuming no further forest loss, and even small populations could persist with immediate protection. Three metapopulations had a moderate/high risk of extirpation, suggesting extinction debt may be evident in parts of the species’ range. When deforestation was assumed to continue at current rates, extirpation risk significantly increased while abundance and genetic diversity decreased for all metapopulations. Extirpation risk was significantly negatively correlated with the size of the largest patch available to metapopulations, underscoring the importance of large habitat patches for species persistence. Finally, we conducted sensitivity analysis using logistic regression, and our results showed that local extinction risk was sensitive to percentage of females breeding, adult female mortality, and dispersal rate and survival; conservation or research programs that target these aspects of the species’ biology/ecology could have a disproportionately important impact on species survival. We stress that efforts to protect populations and tracts of habitat of sufficient size throughout the species’ distribution will be important in the near‐term to protect the species from continuing decline and extinction.     

A Bayesian, combinatorial approach to capture-recapture

It is shown that, in the capture-recapture method, the widely used formulae of Bailey or Chapman-Seber give the most likely value for the size of the population, but systematically underestimate the probability that the population is larger than any given size. We take here a first step in a combinatorial approach which does not suffer from this flaw: formulae are given which can be used in the closed case (no birth, death or migrations between captures) when at least two animals have been recaptured and when there is homogeneity with regard to capture probability. Numerical and heuristic evidence is presented pointing to the fact that the error incurred when using the formulae of Bailey or Chapman-Seber depends asymptotically only on the number of recaptured animals, and will not diminish if the number of captured animals becomes large while the number of recaptured animals remains constant. A result that was stated and left unproven by Darroch is proven here.     

Abundance�Coccupancy relationships in metapopulations: examples of rock pool Daphnia

Intraspecific positive relationships between abundance and occupancy are observed for many species, suggesting that the same processes drive local and regional species dynamics. Two main groups of mechanisms explain this relationship: spatiotemporal variation in local population growth rates due to variation in habitat quality, or dispersal effects that increase occupancy of a species when locally abundant. Several studies show that spatiotemporal variation in population growth rates causes positive abundance?Coccupancy relationships, but few have shown dispersal effects. It is believed that such effects should be more evident for species whose dispersal is limited, e.g. metapopulations, but those studies are limited. This study investigates abundance?Coccupancy relationships in three Daphnia metapopulations in rock pools and the degree to which dispersal or habitat quality affect their local abundances and occurrence. Daphnia longispina and Daphnia magna showed positive abundance?Coccupancy relationships, but not Daphnia pulex. No single ecological factor could explain the abundance?Coccupancy relationships of any given species. Instead, dispersal processes and rock pool quality (mainly salinity and depth) seem to act together to shape the abundance?Coccupancy relationships. Such a conclusion is also supported by an immigration experiment in natural rock pools. This study suggests that although positive abundance?Coccupancy relationships may be commonly found for metapopulations, both dispersal processes and variation in habitat quality can be factors determining the abundance?Coccupancy relationship of metapopulations experiencing habitat heterogeneity.     

Metapopulation vicariance explains old endemics on young volcanic islands

Terrestrial plants and animals on oceanic islands occupy zones of volcanism found at intraplate localities and along island arcs at subduction zones. The organisms often survive as metapopulations, or populations of separate sub‐populations connected by dispersal. Although the individual islands and their local subpopulations are ephemeral and unstable, the ecosystem dynamism enables metapopulations to persist in a region, more or less in situ, for periods of up to tens of millions of years. As well as surviving on systems of young volcanic islands, metapopulations can also evolve there; tectonic changes can break up widespread insular metapopulations and produce endemics restricted to fewer islands or even a single island. These processes explain the presence of old endemic clades on young islands, which is often reported in molecular clock studies, and the many distribution patterns in island life that are spatially correlated with tectonic features. Metapopulations can be ruptured by sea floor subsidence, and this occurs with volcanic loading in zones of active volcanism and with sea floor cooling following its production at mid‐ocean ridges. Metapopulation vicariance will also result if an active zone of volcanism is rifted apart. This can be caused by the migration of an arc (by slab rollback) away from a continent or from another subduction zone, by the offset of an arc at transform faults and by sea floor spreading at mid‐ocean ridges. These mechanisms are illustrated with examples from islands in the Caribbean and the Pacific. Endemism on oceanic islands has usually been attributed to chance, long‐distance dispersal, but the processes discussed here will generate endemism on young volcanic islands by vicariance.     

Probabilistic measures of persistence and extinction in measles (meta)populations

Persistence and extinction are fundamental processes in ecological systems that are difficult to accurately measure due to stochasticity and incomplete observation. Moreover, these processes operate on multiple scales, from individual populations to metapopulations. Here, we examine an extensive new data set of measles case reports and associated demographics in pre‐vaccine era US cities, alongside a classic England & Wales data set. We first infer the per‐population quasi‐continuous distribution of log incidence. We then use stochastic, spatially implicit metapopulation models to explore the frequency of rescue events and apparent extinctions. We show that, unlike critical community size, the inferred distributions account for observational processes, allowing direct comparisons between metapopulations. The inferred distributions scale with population size. We use these scalings to estimate extinction boundary probabilities. We compare these predictions with measurements in individual populations and random aggregates of populations, highlighting the importance of medium‐sized populations in metapopulation persistence.     

Molecular diversity after a range expansion in heterogeneous environments

Recent range expansions have probably occurred in many species, as they often happen after speciation events, after ice ages, or after the introduction of invasive species. While it has been shown that range expansions lead to patterns of molecular diversity distinct from those of a pure demographic expansion, the fact that many species do live in heterogeneous environments has not been taken into account. We develop here a model of range expansion with a spatial heterogeneity of the environment, which is modeled as a gamma distribution of the carrying capacities of the demes. By allowing temporal variation of these carrying capacities, our model becomes a new metapopulation model linking ecological parameters to molecular diversity. We show by extensive simulations that environmental heterogeneity induces a loss of genetic diversity within demes and increases the degree of population differentiation. We find that metapopulations with low average densities are much more affected by environmental heterogeneity than metapopulations with high average densities, which are relatively insensitive to spatial and temporal variations of the environment. Spatial heterogeneity is shown to have a larger impact on genetic diversity than temporal heterogeneity. Overall, temporal heterogeneity and local extinctions are not found to leave any specific signature on molecular diversity that cannot be produced by spatial heterogeneity.